A new class of solvable nonlinear difference equation systems

TL;DR

This paper introduces a new class of nonlinear difference equations and provides a practical method to obtain their solutions in closed form, expanding the set of solvable nonlinear systems.

Contribution

It presents a novel class of nonlinear difference equations and derives their general solutions explicitly, which was not previously available.

Findings

Derived closed-form solutions for the new class of systems

Established a practical method for solving similar nonlinear difference equations

Expanded the understanding of solvability in nonlinear difference systems

Abstract

The paper deals with the following system of nonlinear difference equations \begin{equation*} x_{n+1}=ax_{n}^{2}y_{n}+bx_{n}y_{n}^{2},\ y_{n+1}=cx_{n}^{2}y_{n}+dx_{n}y_{n}^{2},\ n\in \mathbb{N}_{0}, \end{equation*} where the initial values $x_{0},y_{0}$ and the parameters $a$, $b$, $c$, $d$ are arbitrary real numbers, which is a new class of solvable systems of nonlinear difference equations. The general solution of the system is here obtained in closed form via a practical method.